Factorials

Learning Objectives Define the term factorial and use the correct notation (n!). Calculate the value of a factorial for any non-negative integer. Simplify algebraic expressions and fractions involving factorials. Solve simple equations that include factorial terms. Explain the special case of zero factorial (0!). Compare the rapid growth of factorials to linear and quadratic functions. Apply factorials to solve basic counting problems. How many different ways can you arrange 5 books on a shelf? 📚 The answer is surprisingly large and is found using a special operation called a factorial! In this tutorial, you will learn about factorials, a powerful mathematical operation used for counting arrangements. While we've studied linear, quadratic, and exponential growth, fa...

TermDefinitionExample FactorialThe factorial of a non-negative integer 'n', denoted as n!, is the product of all positive integers less than or equal to n.5! = 5 × 4 × 3 × 2 × 1 = 120 Factorial NotationThe symbol '!' placed after a non-negative integer indicates that you should calculate its factorial.The expression '4!' is read as 'four factorial'. Non-negative IntegersThe set of numbers for which factorials are defined. This includes zero and all positive whole numbers.{0, 1, 2, 3, 4, ...} Zero FactorialA special case by definition where the factorial of zero is equal to 1. This is important for many mathematical formulas to work correctly.0! = 1 Recursive DefinitionA way of defining a concept in terms of itself. A factorial can be defined recursi...

Definition of a Factorial For a positive integer n, n! = n × (n-1) × (n-2) × ... × 2 × 1 This is the fundamental formula used to calculate the value of any factorial. You multiply the number by every positive integer smaller than it. The Zero Factorial Rule 0! = 1 This is a specific rule that must be memorized. It defines the value of zero factorial as 1. The Recursive Rule n! = n × (n-1)! for n ≥ 1 This rule is very useful for simplifying expressions. It shows that any factorial is simply the number 'n' multiplied by the factorial of the number just before it. Factorial Division Rule n! / k! = n × (n-1) × ... × (k+1) for n > k When dividing a larger factorial by a smaller one, you can cancel out the common terms, leaving a product of the integer...